Restriction of toral eigenfunctions to totally geodesic submanifolds

Huang, Xiaoqi; Zhang, Cheng

doi:10.2140/apde.2021.14.861

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Restriction of toral eigenfunctions to totally geodesic submanifolds

Xiaoqi Huang and Cheng Zhang

Vol. 14 (2021), No. 3, 861–880

DOI: 10.2140/apde.2021.14.861

Abstract

We estimate the $L^{2}$ norm of the restriction to a totally geodesic submanifold of the eigenfunctions of the Laplace–Beltrami operator on the standard flat torus $𝕋^{d}$ , $d \geq 2$ . We reduce getting correct bounds to counting lattice points in the intersection of some $ν$ -transverse bands on the sphere. Moreover, we prove the correct bounds for rational totally geodesic submanifolds of arbitrary codimension. In particular, we verify the conjecture of Bourgain–Rudnick on $L^{2}$ -restriction estimates for rational hyperplanes. On $𝕋^{2}$ , we prove the uniform $L^{2}$ restriction bounds for closed geodesics. On $𝕋^{3}$ , we obtain explicit $L^{2}$ restriction estimates for the totally geodesic submanifolds, which improve the corresponding results by Burq–Gérard–Tzvetkov, Hu, and Chen–Sogge.

Keywords

eigenfunction estimates, restriction, geodesic

Mathematical Subject Classification 2010

Primary: 11P21, 35P20, 58J50

Milestones

Received: 27 February 2019

Revised: 24 September 2019

Accepted: 21 November 2019

Published: 18 May 2021

Authors

Xiaoqi Huang
Department of Mathematics Johns Hopkins University Baltimore, MD United States

Cheng Zhang
Department of Mathematics Johns Hopkins University Baltimore, MD United States

Vol. 14, No. 3, 2021